Ellipse of Hooke

The ellipse of Hooke is the trajectory of a mobile elastically related to a fixed point.

History

Robert Hooke (1635 - 1703) stated his “Loi of Hooke” on the Ressort S in 1670:

F = - K ·Δ L

F being the force exerted by the spring, Δ L its lengthening and K the constant of stiffness. It understood that in the case of the spherical Pendule (a small heavy mass attached by a string length L fixed in a point O ), if one considered only the small oscillations, the force of recall towards the position of steady balance (the downward vertical), composed of the weight and the tension of the wire, was written

$\backslash \; vec\; \{F\}\; (M)\; =\; -\; K\; \backslash \; cdot\; \backslash \; overrightarrow\; \{OM\}$,

with K = m ·( G / L ), G being the acceleration of the Revolved.

The Theorem of the kinetic energy of Torricelli moulted itself in this case in:

$\backslash \; frac\; \{1\}\; \{2\}\; m\; \backslash \; left\; (v\_x^2\; +\; v\_y^2\; \backslash \; right)\; +\; \backslash \; frac\; \{1\}\; \{2\}\; K\; \backslash \; cdot\; \backslash \; left\; (x^2\; +\; y^2\; \backslash \; right)\; =\; E\_0$

(the kinetic energy 1/2· m · v ² varies contrary to elastic potential energy 1/2· K · OM ²), where

• X and is there the Cartesian coordinates of the mobile M compared to the point of fastener O ;
• v_x and v_y is the component of the vector Speed.

This differential equation of Newton, as we say nowadays, had already been solved the case with a dimension (linear movement) by Huygens in its Horologium ; Huygens had found the solution intuitively:

$x\; (T)\; =\; has\; \backslash \; cdot\; \backslash \; cos\; (\backslash \; Omega\; \backslash \; cdot\; T\; +\; \backslash \; varphi)$

thus finding the formula of the simple pendulum of Galileo:

$T\; =\; 2\; \backslash \; cdot\; \backslash \; pi\; \backslash \; cdot\; \backslash \; sqrt\; \{\backslash \; frac\; \{L\}\; \{G\}\}$.

Robert Hooke simply diluted this solution to the case with two dimensions (plane movement): he noticed that by taking X ( T ) and there ( T ) sinusoidal of the same pulsation, the movement was a ellipse (known as of Lissajous in France). Was its knowledge in “calculus” quasi-non-existent (? ).

Newton, wrote to him (almost like that):

$x$ + \ omega_0^2 \ cdot X =0

- and -

$y$ + \ omega_0^2 \ cdot there =0

thus

$\backslash \; overrightarrow\; \{OM\}\; =\; \backslash \; overrightarrow\; \{OM\_0\}\; \backslash \; cdot\; \backslash \; cos\; (\backslash \; omega\_0\; T)\; +\; \backslash \; frac\; \{\backslash \; vec\; \{V\_0\}\}\; \{\backslash \; omega\_0\}\; \backslash \; cdot\; \backslash \; sin\; (\backslash \; omega\_0\; T)$

- and -

Hodograph $\backslash \; vec\; \{V\}\; (T)$ invariant by the temporal translation T → T + T /4

what is the definition of an ellipse by its combined diameters, well-known in projective geometry (see circle in axonometric Perspective): construction is done easily with the rule and the compass. (promised diagram as soon as I can draw) .

The theorems of Apollonius find a simple physical interpretation there (energy E ₀, kinetic Moment L ₀). Moreover, there exists a tensorial invariant additional, to which in quantum Mécanique corresponds a degeneration of the energy levels (cf Symétrie).

In the case of a homogeneous ball, the internal field of gravity is

$-\; K\; \backslash \; cdot\; \backslash \; overrightarrow\; \{OM\}$

with

K = m · G / R

so that the internal trajectory of a small mass would be an ellipse of Hooke of period

$T\; =\; 2\; \backslash \; pi\; \backslash \; cdot\; \backslash \; sqrt\; \{\backslash \; frac\; \{R\}\; \{G\}\}$,

who is thus independent of the ray of the ball, but only of his density, result which does not astonish that which knows the theorem of Newton-Gauss on the hollow hulls:

the mass beyond the apogee of the trajectory is without action of gravity .

Did Hooke have intuity this theorem before Newton? The destruction in 1703 of the files of Hooke by Newton to the Royal Society is well character of Newton, but leaves little chance to be able to conclude.

The model of Thomson known as of the elastically bound electron

In atomic physics, one uses by convenience the model of Thomson: a electron would move in a uniform ball of opposite load of ray approximately 0,1nm. Obviously, one can also say that it is the spherical envelope of the “delocalized” electron which moves in block compared to the quasi-specific core (one in any event utilizes the reduced Masse). This model has the good taste to adapt to grinds explanations in atomic physics, and thus is very much used in teaching.

It is thus the exact transposition in electricity of a writing of Hooke of 1679.

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